The basis for phononic crystals-dates back to Isaac Newton who imagined that sound waves propagated through air in the same way that an elastic wave would propagate along a lattice of point masses connected by springs with an elastic force constant E, the force constant being identical to the modulus of the material. The field of phononic crystals and our theoretical understanding of them have steadily grown since that time (see, for example, Joannopoulos, R. D. Meade and J. N. Winn, Photonic Crystals, Molding the Flow of Light (Princeton University Press, Princeton (1995); Garcia et al., “Theory for Tailoring Sonic Devices: Diffraction Dominates over Refraction,” Phys. Rev. E 67, 046606 (2003); Kushwaha and P. Halevi, “Band-gap Engineering in Periodic Elastic Composites,” Appl. Phys. Lett. 64(9):1085-1087 (1994); Lai et al. “Engineering Acoustic Band Gaps,” Appl. Phys. Lett. 79(20): 3224-3226 (2001); Sigmund and Jensen “Systematic Design of Phononic Band-Gap Materials and Structures by Topology Optimization,” Phil. Trans. R. Soc. Lond. A 361:1001-1019 (2003), Caballero et al. “Large Two-Dimensional Sonic Band Gaps,” Phys. Rev. E 60:(6):R6316-R6319 (1999); and Sliwa and Krawczyk “The Effect of Material Parameters Values on the Relation Between Energy Gap Width and the Scattering Symmetry in Two-Dimensional Phononic Crystals,” arXiv:cond-mat/05022 (2005).